Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 28, 2003, 35–54
A GENERALIZED NEVANLINNA
THEOREM FOR SUPERTEMPERATURES
Neil A. Watson
University of Canterbury, Department of Mathematics and Statistics
Christchurch, New Zealand; N.Watson@math.canterbury.ac.nz
Abstract. A new proof of Nevanlinna’s first fundamental theorem for supertemperatures
enables us to generalize the result to cover the case of the mean values of supertemperatures over
level surfaces of Green functions generally. Generalizations in another direction are also obtained.
1. Introduction
In [11], an analogue for supertemperatures of Nevanlinna’s First Fundamental
Theorem for superharmonic functions was presented. A new and much simpler
proof has now been discovered, which enables us to generalize the theorem to
include the general case of the mean values of supertemperatures over level surfaces
of Green functions considered in [10]. This is the result of Theorem 1 below, and it
is followed by an application to thermic majorization in Theorem 2. Subsequently,
all the theorems in [11] are generalized in this way, and those which are concerned
with supertemperatures on lower half-spaces Rn ×] − ∞, a[ are extended to results
on sets of the form Λ(p0 , D) which appear in the statement of the strong minimum
principle, P
where D is any open set that is Dirichlet regular for the adjoint heat
n
operator i=1 Di2 + Dt .
Generalizations in another direction are also obtained. Instead of the quotients of surface mean values that appear in [11, Theorem 2], we use quotients
of differences of the generalized mean values below. This renders the condition
“ v(p0 ) = ∞ ” redundant, and the corresponding infinity conditions of [11, Theorems 3 and 4] are also absent from the generalizations below. One consequence of
this is that, whereas [11, Theorem 4] gave Hausdorff measure estimates of certain
polar sets, in the corresponding result below the sets need not be polar.
We work in Rn+1 , a typical point of which we usually denote by p or q , and
rarely by (x, t) with x ∈ Rn and t ∈ R . We denote by D an open subset of
Rn+1 that is Dirichlet regular for
Pnthe adjoint heat operator, and by G D its Green
function for the heat operator i=1 Di2 − Dt (but in the case D = Rn+1 we omit
the subscript). All our positive measures are locally finite Borel measures, and our
signed measures are differences of pairs of positive measures. The terms ‘positive’
2000 Mathematics Subject Classification: Primary 31C05, 35K05.
36
Neil A. Watson
and ‘increasing’ are used in the wide sense. Given a positive measure µ on D , its
Green potential GD µ is defined by
Z
GD µ(p) =
GD (p, q) dµ(q)
D
for all p ∈ D . A temperature is a solution of the heat equation, and a supertemperature is the corresponding analogue of a superharmonic function. A thermic
minorant of a supertemperature w is a temperature u such that u ≤ w . We
denote by E an open subset of Rn+1 (that is not necessarily Dirichlet regular). If
w is a supertemperature on E , then the Riesz Decomposition Theorem associates
with w a positive measure on E , which we call the Riesz measure for w . (See [6]
and [7], or [3], for details.) If v is also a supertemperature on E , and u = v − w
whenever the difference is defined (hence on E less a polar set), then u is called
a δ -subtemperature on E . If ν and ω are the Riesz measures for v and w respectively, then ν − ω will be called the Riesz (signed) measure for u . The Riesz
measure for u is uniquely determined [11].
For all c > 0 , we put τ (c) = (4πc)−n/2 . Let p0 ∈ D . There is a positive,
bounded solution h of the adjoint heat equation on D such that GD (p0 , · ) =
G(p0 , · ) − h , so that GD (p0 , · ) is infinitely differentiable on D\{p0 } . It therefore
follows from Sard’s Theorem ([4, p. 45]) that, for almost every c > 0 , the set
{p ∈ D : GD (p0 , p) = τ (c)} is a smooth regular n -dimensional manifold. We call
such a value of c a regular value. We put
©
ª
ΩD (p0 , c) = p ∈ D : GD (p0 , p) > τ (c)
and
0
ΩD (p0 , c) = ΩD (p0 , c)\{p0 },
omitting the subscript if D = Rn+1 (in which case Ω(p0 , c) is the heat ball with
centre p0 and radius c ). Because GD (p0 , · ) is a solution of the adjoint equation
on D\{p0 } , each set ΩD (p0 , c) is Dirichlet regular for that equation. For any
regular value of c ,
©
ª
∂ΩD (p0 , c) = p ∈ D : GD (p0 , p) = τ (c) ∪ {p0 }.
Since GD (p0 , · ) is lower semicontinuous on D , each ΩD (p0 , c) is an open set.
The assumption that D is Dirichlet regular for the adjoint operator implies that
GD (p0 , · ) can be continuously extended to zero on ∂D , so that ΩD (p0 , c) ⊆ D
for all c > 0 . Furthermore, ΩD (p0 , c) is bounded and connected ([10, p. 167]).
If c is a regular value, then the outward unit normal ν = (νx , νt ) to ∂ΩD (p0 , c)
is given by the standard formula ν = −∇GD (p0 , · )k∇GD (p0 , · )k−1 . Therefore, if
∇x u = (D1 u, . . . , Dn u) denotes the gradient in the spatial variables only, we have
h∇x GD (p0 , · ), νx i = −k∇x GD (p0 , · )k2 k∇GD (p0 , · )k−1 .
A generalized Nevanlinna theorem for supertemperatures
37
This function is bounded on ∂ΩD (p0 , c)\{p0 } (see [10, pp. 167–8]). The surface
mean value over ∂ΩD (p0 , c) is defined by
MD (u, p0 , c) =
Z
∂ΩD (p0 ,c)
−h∇x GD (p0 , · ), νx iu dσ
whenever the integral exists; here σ denotes surface area measure.
2. The generalized Nevanlinna theorem
In this section we present our generalization of [11, Theorem 1], along with
some immediate consequences.
Theorem 1. Let E be an open set, let D be an open superset of E that
is Dirichlet regular for the adjoint heat operator, let p0 ∈ E , and let c and d be
regular values such that 0 < c ≤ d and ΩD (p0 , d) ⊆ E . If w is a supertemperature
on E with Riesz measure µ , then
(1)
MD (w, p0 , c) = MD (w, p0 , d) −
Z
d
c
¢
¡ 0
τ 0 (γ)µ ΩD (p0 , γ) dγ
and
(2)
w(p0 ) = MD (w, p0 , d) −
Z
d
0
¡ 0
¢
τ 0 (γ)µ ΩD (p0 , γ) dγ.
Proof. Let V be a bounded open set such that ΩD (p0 , d) ⊆ V and V ⊆ E .
By adding a constant if necessary, we may suppose that w ≥ 0 on V . Then w
can be written in the form w = GµV + u on V , where µV is the restriction of µ
to V extended by zero to Rn+1 . Since µV is finite, GµV is a supertemperature
on Rn+1 . By [7, Theorem 19], there is a temperature v on V such that GµV =
GD µV +v on D , so that w = GD µV +h on V , where h = u+v . Now observe that
the means in (1) are finite (by [10, Theorem 2], or [1], [2]), and that M D (h, p0 , c) =
h(p0 ) (by [10, Theorem 1]). It follows that
MD (w, p0 , c) − MD (w, p0 , d) = MD (GD µV , p0 , c) − MD (GD µV , p0 , d)
Z ³
¡
¢
=
MD GD ( · , q), p0 , c
V
¡
¢´
− MD GD ( · , q), p0 , d dµ(q)
Z
¡
¢ ¡
¢¢
(τ (c) ∧ GD (p0 , q) − τ (d) ∧ GD (p0 , q) dµ(q)
=
V
38
Neil A. Watson
in view of [2, Theorem 2 Corollary], and [1]. By definition of Ω D (p0 , c) , we have
0
τ (c) ∧ GD (p0 , q) = τ (c) if and only if q ∈ ΩD (p0 , c) , so that
¡
¢ ¡
¢
τ (c) ∧ GD (p0 , q) − τ (d) ∧ GD (p0 , q)

0

if q ∈ ΩD (p0 , c),
 τ (c) − τ (d)
= GD (p0 , q) − τ (d) if q ∈ ΩD (p0 , d)\ΩD (p0 , c),


0
0
if q ∈
/ ΩD (p0 , d).
Hence
MD (w, p0 , c) − MD (w, p0 , d) =
Z
ΩD (p0 ,d)
¡¡
¢
¢
τ (c) ∧ GD (p0 , q) − τ (d) dµ(q).
¡ 0
¢
If we now put λ(γ) = µ ΩD (p0 , γ) whenever 0 < γ ≤ d , we have
MD (w, p0 , c) − MD (w, p0 , d) =
Z
0
d ¡¡
¢
¢
τ (c) ∧ τ (γ) − τ (d) dλ(γ)
¯d Z d
¯
= τ (c) ∧ τ (γ) − τ (d) λ(γ) ¯ −
τ 0 (γ)λ(γ) dγ
0
c
Z d
¢
¡ 0
τ 0 (γ)µ ΩD (p0 , γ) dγ,
=−
¡¡
¢
¢
c
which proves (1). Making c → 0 in (1), we obtain (2).
The corollaries of [11, Theorem 1] can also be extended to the present situation.
Corollary 1. Let E be an open set, let D be an open superset of E that
is Dirichlet regular for the adjoint heat operator,
¡ and
¢ let p0 ∈ E . If w is a
supertemperature on E , then MD (w, p0 , c) = o τ (c) as c → 0 through regular
values.
The proof is similar to that of the case D = Rn+1 in [11].
Corollary 2. Let D be an open set which is Dirichlet regular for the adjoint
heat operator, let p0 ∈ D , let w be a positive supertemperature on an open
superset of Λ(p0 , D) ∪ {p0 } , and let µ be the Riesz measure for w . Then
¡ 0
¢
τ (c)µ ΩD (p0 , c) ≤ MD (w, p0 , c)
for all regular values of c .
A generalized Nevanlinna theorem for supertemperatures
39
Proof. Note that GD (p0 , q) > 0 if and only if q ∈ Λ(p0 , D) (see [7, Theorem 14], and [3, p. 300]). Therefore ΩD (p0 , c) ⊆ Λ(p0 , D) ∪ {p0 } for all c > 0 . It
now follows from Theorem 1 that, if c and d are regular values with c < d , then
Z d
¢
¡ 0
MD (w, p0 , c) ≥ −
τ 0 (γ)µ ΩD (p0 , γ) dγ
c
¢¡
¢
¡ 0
≥ µ ΩD (p0 , c) τ (c) − τ (d)
¡ 0
¢
→ τ (c)µ ΩD (p0 , c)
as d → ∞ .
Corollary 3. Let D be an open set which is Dirichlet regular for the adjoint heat operator, let p0 ∈ D , and let µ be the Riesz measure for a positive
supertemperature w on E ∈ {D, Λ(p0 D)} . Then
¢
¡ 0
lim τ (c)µ ΩD (p, c) = 0
c→∞
for all p ∈ E .
Proof. By a result in [9], the domain Λ(p0 , D) is Dirichlet regular for the
adjoint heat operator. Furthermore, the Green function for Λ(p0 , D) is the restriction of GD to Λ(p0 , D) × Λ(p0 , D) (by [7, Theorem 14], and [3, p. 300]), so
that ΩΛ(p0 ,D) (p, c) = ΩD (p, c) for all p ∈ Λ(p0 , D) . It therefore suffices to prove
the result with E = D . The greatest thermic minorant u of w is given by
u(p) = lim MD (w, p, c)
c→∞
for all p ∈ D (see [10, Theorem 7], and [8]). Since µ is also the Riesz measure for
w − u , it therefore follows from Corollary 2 and [10, Theorem 1] that
¢
¡ 0
τ (c)µ ΩD (p0 , c) ≤ MD (w − u, p, c) = MD (w, p, c) − u(p) → 0
as c → ∞ through regular values. Hence, given ε > 0 we can find K such that
¡ 0
¢
µ ΩD (p0 , c) ≤ ετ (c)−1 for all regular values of c > K , and hence for all c > K
¢
¡ 0
because µ ΩD (p0 , · ) is an increasing function of c , and τ is continuous.
Theorem 1 can also be used to extend [10, Theorem 7].
Theorem 2. Let D be an open set which is Dirichlet regular for the adjoint heat operator, and let w be a supertemperature on D . Then the following
statements are equivalent:
(i) w has a thermic minorant on D .
S∞
(ii) There is a sequence {pj } in D such that D = j=1 Λ(pj , D) and for each j
the function MD (w, pj , · ) is bounded below onSthe set of all regular values.
∞
(iii) There is a sequence {pj } in D such that D = j=1 Λ(pj , D) and
Z ∞
¢
¡ 0
(3)
γ −(n+2)/2 µ ΩD (pj , γ) dγ < ∞
1
for all j , where µ is the Riesz measure for w .
40
Neil A. Watson
Proof. The equivalence of (i) and (ii) is [10, Theorem 7]. To S
prove that (ii)
∞
and (iii) are equivalent, let {pj } be a sequence in D such that D = j=1 Λ(pj , D) .
Given j , if c and d are regular values and c < d , then
Z d
¢
¡ 0
0
MD (w, pj , c) = MD (w, pj , d) − τ (1)
γ −(n+2)/2 µ ΩD (pj , γ) dγ
c
by Theorem 1. Furthermore, the means are finite valued ([10, Theorem 2], or [2]).
Therefore, if we fix c and make d → ∞ , we see that MD (w, pj , · ) is bounded
below if and only if (3) holds.
3. The behaviour of the means for small regular values
Theorem 3 below generalizes [11, Theorem 2] in two directions. First it removes the restriction v(p0 ) = ∞ , and second it replaces M by MD .
We need some notation. Let E be an open set, and let D be an open superset
of E that is Dirichlet regular for the adjoint heat operator. If ΩD (p0 , d) ⊆ E , ν
is a positive measure on E , and 0 ≤ b < c ≤ d , we put
Z c
Z c
¢
¢
¡ 0
¡ 0
0
Iν,D (p0 ; b, c) = −
τ (γ)ν ΩD (p0 , γ) dγ = κn
γ −(n+2)/2 ν ΩD (p0 , γ) dγ,
b
b
where κn = −τ 0 (1) = n2−n−1 π −n/2 .
We include for completeness the definition of
lim sup f (b, c),
0<b<c→0
although it is the natural one. Those of the corresponding lim inf and lim are
then obvious.
Definition. Suppose that f (b, c) is defined as an extended-real number for
Lebesgue almost all b and c such that 0 < b < c < d , and that l ∈ R . We write
lim sup f (b, c) = l
0<b<c→0
if to each ε > 0 there corresponds δ > 0 such that f (b, c) < l + ε whenever
f (b, c) is defined with 0 < b < c < δ , and there is a sequence {(bk , ck )} such that
0 < bk < ck → 0 and f (bk , ck ) → l as k → ∞ . We also write
lim sup f (b, c) = ∞
0<b<c→0
if there is a sequence {(bk , ck )} such that 0 < bk < ck → 0 and f (bk , ck ) → ∞ .
Finally, we write
lim sup f (b, c) = −∞
0<b<c→0
if to each A ∈ R there corresponds δ > 0 such that f (b, c) < A whenever f (b, c)
is defined with 0 < b < c < δ .
A generalized Nevanlinna theorem for supertemperatures
41
Theorem 3. Let E be an open set, let D be an open superset of E that is
Dirichlet regular for the adjoint heat operator, let u be a δ -subtemperature on E
with Riesz measure µ , and let ν be a positive measure on E . Then
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
≤ lim sup ¡ 0
(4)
lim sup
¢
Iν,D (p; b, c)
d→0 ν ΩD (p, d)
0<b<c→0
whenever the latter exists. Furthermore, if Iν,D (p; 0, c) < ∞ for all sufficiently
small values of c , and u(p) is defined and finite, then
¢
¡ 0
µ ΩD (p, d)
u(p) − MD (u, p, c)
≤ lim sup ¡ 0
(5)
lim sup
¢.
Iν,D (p; 0, c)
c→0
d→0 ν ΩD (p, d)
Proof. Suppose that the upper limit on the right-hand side of (4) exists, and
denote it by l . If l = ∞ there is nothing to prove. Otherwise, given a real number
A > l , we can find δ > 0 such that
¡ 0
¢
µ ΩD (p, d)
(6)
whenever 0 < d < δ.
¢ <A
¡ 0
ν ΩD (p, d)
¢
¡ 0
¢
¡ 0
If ν ΩD (p, d) = 0 for all d < η(≤ δ) , then (6) can hold only if µ ΩD (p, d) < 0
for all d < η . Then Iν,D (p; b, c) = 0 whenever c < η , and (1) shows that
MD (u, p, b) − MD (u, p, c) < 0
for all regular values such that 0 < b < c < η , so that (4) holds with both
¡ 0
¢
sides −∞ . On the other hand, if ν ΩD (p, d) > 0 for all d , then by (1)
¡ 0
¢
Z c
¡
¢
Ω
µ
(p,
γ)
−1
MD (u, p, b) − MD (u, p, c)
0
=
τ 0 (γ)ν ΩD (p, γ) ¡ D
¢ dγ
0
Iν,D (p; b, c)
Iν,D (p; b, c) b
ν ΩD (p, γ)
<A
for all regular values such that 0 < b < c < δ , and again (4) holds.
The inequality (5) can be proved in a similar way, using (2) instead of (1).
Corollary. If u is a δ -subtemperature with Riesz measure µ on an open set
E , then
¡ 0
¢
µ Ω (p, d)
M (u, p, b) − M (u, p, c)
(7)
lim
= κn lim
0<b<c→0
d→0 d(n+2)/2
c−b
whenever the latter exists. Furthermore, if u(p) is defined and finite, then
¡ 0
¢
µ Ω (p, d)
M (u, p, c) − u(p)
lim
= −κn lim
c→0
d→0 d(n+2)/2
c
whenever the latter exists.
42
Neil A. Watson
Proof. We take D = Rn+1 and ν to be (n+1) -dimensional Lebesgue measure
in Theorem 3, noting that the reverse inequalities hold for lower limits. A routine
¢
¡ 0
calculation shows that ν Ω (p, γ) = λn γ (n+2)/2 , where
λn = 2n+1 (nπ)n/2 (n + 2)−(n+2)/2 .
Therefore, if 0 ≤ b < c and Ω(p, c) ⊆ E , then
Iν (p; b, c) = κn λn
Z
c
dγ =
b
µ
n
n+2
¶(n+2)/2
(c − b).
It now follows from (4) and (5), together with their duals for lower limits, that
¡ 0
¢
µ Ω (p, d)
M (u, p, b) − M (u, p, c)
= lim
lim
d→0 λn d(n+2)/2
0<b<c→0
κn λn (c − b)
and
¢
¡ 0
µ Ω (p, d)
u(p) − M (u, p, c)
lim
= lim
c→0
d→0 λn d(n+2)/2
κ n λn c
whenever the last limit exists.
It is not immediately apparent that Theorem 3 is a generalization of [11,
Theorem 2]. To demonstrate that it is, we first write it in a different form, then
deduce the earlier result as the case D = Rn+1 of the subsequent corollary.
Theorem 4. Let E be an open set, and let D be an open superset of E
that is Dirichlet regular for the adjoint operator. Let u be a δ -subtemperature
with Riesz measure µ , and v a supertemperature with Riesz measure ν , on E .
Then
(8)
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
lim sup
≤ lim sup ¡ 0
¢
0<b<c→0 MD (v, p, b) − MD (v, p, c)
d→0 ν ΩD (p, d)
whenever the latter exists. Furthermore, if u(p) is defined and finite, and v(p) <
∞ , then
¡ 0
¢
µ ΩD (p, d)
u(p) − MD (u, p, c)
lim sup
≤ lim sup ¡ 0
¢.
v(p) − MD (v, p, c)
c→0
d→0 ν ΩD (p, d)
Proof. In view of Theorem 1 and the finiteness of the mean values, the result
follows from Theorem 3.
A generalized Nevanlinna theorem for supertemperatures
43
Corollary. If the hypotheses of Theorem 4 are satisfied and v(p) = ∞ , then
¡ 0
¢
µ ΩD (p, c)
MD (u, p, c)
≤ lim sup ¡ 0
lim sup
¢.
MD (v, p, c)
c→0
c→0
ν ΩD (p, c)
(9)
Proof. Let l denote the left-hand side of (8). We prove that
(10)
lim sup
c→0
MD (u, p, c)
≤ l,
MD (v, p, c)
which implies that (9) holds, in view of (8). We may assume that l < ∞ . Given
a real number A > l , we choose δ > 0 such that
MD (u, p, b) − MD (u, p, c)
<A
MD (v, p, b) − MD (v, p, c)
whenever b and c are regular values such that 0 < b < c < δ . By [10, Theorem 2],
MD (v, p, d) → v(p) as d → 0 (through regular values), so that our hypothesis
v(p) = ∞ means we may suppose that MD (v, p, d) > 0 for all regular values of
d < δ . With this assumption, we fix a regular value of c < δ . Given ε > 0 , we
choose η < c such that both
MD (v, p, c)
<ε
MD (v, p, b)
and
MD (u, p, c)
<ε
MD (v, p, b)
for all regular values of b < η . Then
µ
¶
MD (u, p, b) − MD (u, p, c)
MD (v, p, c)
MD (u, p, c)
MD (u, p, b)
=
1−
+
MD (v, p, b)
MD (v, p, b) − MD (v, p, c)
MD (v, p, b)
MD (v, p, b)
©
ª
< max A, (1 − ε)A + ε
if b < η . This proves (10), and (9) follows.
We can also generalize [11, Theorem 3] in a similar way.
Theorem 5. Let E be an open set, let D be an open superset of E that is
Dirichlet regular for the adjoint operator, and let u be a δ -subtemperature with
Riesz measure µ on E . Let α > 0 , let f be a positive, increasing, absolutely
continuous function on [0, α] , and let
fˆ(b, c) = κn
Z
c
b
γ −(n+2)/2 f (γ) dγ
44
Neil A. Watson
whenever 0 ≤ b < c ≤ α . Then
(11)
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
≤ lim sup
lim sup
f (d)
d→0
0<b<c→0
fˆ(b, c)
for every p ∈ E . Furthermore, if u(p) is defined and finite, and fˆ(0, c) < ∞ for
all sufficiently small values of c , then
(12)
¡ 0
¢
µ ΩD (p, d)
u(p) − MD (u, p, c)
≤ lim sup
lim sup
.
f (d)
c→0
d→0
fˆ(0, c)
¡ 0
¢
¡
¢
Proof. If f (0) 6= 0 , then µ ΩD (p, d) = o f (d) as d → 0 , so we have to
prove that
MD (u, p, b) − MD (u, p, c)
= 0.
lim sup
0<b<c→0
fˆ(b, c)
(Note that in this case fˆ(0, c) = ∞ , so that the conditions for (12) are not satisfied.) If 0 < b < c < α , then
fˆ(b, c) ≥ f (0)
Z
c
b
so that
0≤
¡
¢
−τ 0 (γ) dγ = f (0) τ (b) − τ (c) ,
τ (b) − τ (c)
1
.
≤
f (0)
fˆ(b, c)
Also, by Theorem 1,
MD (u, p, b) − MD (u, p, c)
1
=
τ (b) − τ (c)
τ (b) − τ (c)
Z
c
b
¡ 0
¢
−τ 0 (γ)µ ΩD (p, γ) dγ,
¢
¡ 0
which is o(1) as 0 < b < c → 0 because µ ΩD (p, γ) = o(1) as γ → 0 . It follows
that
µ
¶
¶µ
MD (u, p, b) − MD (u, p, c)
MD (u, p, b) − MD (u, p, c)
τ (b) − τ (c)
=
= o(1)
τ (b) − τ (c)
fˆ(b, c)
fˆ(b, c)
as 0 < b < c → 0 through regular values.
Now consider the case where f (0) = 0 . We choose d ≤ α such that ΩD (p, d)
is contained in E , and define a measure ν on E by putting
dν = −k∇x GD (p, · )k
2
µ
f0
τ0
¶µ
¶
GD (p, · )−2/n
χΩD (p,d) dλ,
4π
A generalized Nevanlinna theorem for supertemperatures
45
where χA denotes the characteristic function of a set A , and λ denotes (n + 1) dimensional Lebesgue measure. If 0 < c < d , it follows from results in [10,
pp. 167–70] that
¶
µ 0 ¶µ
Z
¡ 0
¢
GD (p, · )−2/n
2 f
ν ΩD (p, c) = −
dλ
k∇x GD (p, · )k
τ0
4π
ΩD (p,c)
¶
Z c µZ
k∇x GD (p, · )k2 0
=
f (γ) dσ dγ
0
∂ΩD (p,γ) k∇GD (p, · )k
Z c
MD (1, p, γ)f 0 (γ) dγ
=
=
Z
0
c
f 0 (γ) dγ = f (c),
0
so that
Iν,D (p; b, c) = −
Z
c
τ 0 (γ)f (γ) dγ = fˆ(b, c)
b
whenever 0 ≤ b < c ≤ d . The inequalities (11) and (12) now follow from Theorem 3.
The special case D = Rn+1 of the following corollary is [11, Theorem 3].
then
Corollary 1. If the hypotheses of Theorem 5 are satisfied and fˆ(0, α) = ∞ ,
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b)
≤ lim sup
lim sup
f (d)
d→0
b→0
fˆ(b, α)
for each p ∈ E .
Proof. Given p ∈ E , let l denote the left-hand side of (11). In view of (11),
it is more than enough to prove that
(13)
lim sup
b→0
MD (u, p, b)
≤ l.
fˆ(b, α)
We may assume that l < ∞ . Given any real number A > l , we can find δ > 0
such that
MD (u, p, b) − MD (u, p, c)
<A
fˆ(b, c)
whenever b and c are regular values such that 0 < b < c < δ . Fix c < δ . Given
ε > 0 , we choose η < c such that both
MD (u, p, c)
<ε
fˆ(b, α)
and
fˆ(c, α)
<ε
fˆ(b, α)
46
Neil A. Watson
whenever 0 < b < η . Then
µ
¶
MD (u, p, b) − MD (u, p, c)
fˆ(c, α)
MD (u, p, c)
MD (u, p, b)
=
1−
+
fˆ(b, α)
fˆ(b, c)
fˆ(b, α)
fˆ(b, α)
©
ª
< max A, (1 − ε)A + ε
for every regular value of b < η . The inequality (13) follows.
The extra generality of Theorem 5 over its first corollary enables us to generalize [11, Theorem 3 Corollary], and remove its restriction on the range of values
of its parameter β , as follows.
Corollary 2. Let E be an open set, let D be an open superset of E that is
Dirichlet regular for the adjoint operator, let u be a δ -subtemperature with Riesz
measure µ on E , and let p ∈ E . Then
µ
n−β
2
¶
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
lim sup
≤ κn lim sup
b−(n−β)/2 − c−(n−β)/2
dβ/2
0<b<c→0
d→0
if 0 ≤ β < n ,
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
,
≤ κn lim sup
lim sup
log(c/b)
dn/2
d→0
0<b<c→0
and
µ
β−n
2
¶
¡ 0
¢
µ ΩD (p, d)
MD (u, p, b) − MD (u, p, c)
≤ κn lim sup
lim sup
c(β−n)/2 − b(β−n)/2
dβ/2
d→0
0<b<c→0
if β > n .
Proof. We take f (d) = dβ/2 for β ≥ 0 , in Theorem 5. Then
Z c
ˆ
f (b, c) = κn
γ (β−n−2)/2 dγ,
b
which is equal to κn times
2(b−(n−β)/2 − c−(n−β)/2 )/(n − β)
if 0 ≤ β < n,
log(c/b)
if β = n,
2(c(β−n)/2 − b(β−n)/2 )/(β − n)
if β > n,
and the result follows.
A generalized Nevanlinna theorem for supertemperatures
47
4. The parabolic Hausdorff measures of certain sets
We use Theorem 5 to study the size of the sets of points p where
MD (u, p, b) − MD (u, p, c)
fˆ(b, c)
is unbounded as 0 < b < c → 0 , for a given function f and supertemperature u .
The size is estimated in terms of the parabolic Hausdorff measures discussed in
[5], which have the appropriate mixed homogeneity.
We recall the necessary definitions. Let h be an increasing function on ]0, ∞[
such that h(0+) = 0 . Let P denote the class of all sets of the form
µY
¶
n
[ai , ai + r] × [a, a + r 2 ].
i=1
The set of this form which is centred at p is denoted by P (p, r) . For an arbitrary
set S , the outer parabolic h -measure of S is defined by
½X
¾
∞
∞
S
h(diam Pi ) : Pi ∈ P, E ⊆
P − h − m∗ (S) = lim inf
Pi , diam Pi < δ .
δ→0+
i=1
i=1
The associated measure, defined on a σ -field that contains the Borel sets, is denoted by P − h − m . When h(s) = sα for some α > 0 , we write P − Λα − m
for P − h − m .
Theorem 6. Let E be an open set, let D be an open superset of E that is
Dirichlet regular for the adjoint operator, and let u be a supertemperature on E .
Let 0 < α < ∞£ , and
√ ¤let h be an increasing function on [0, ∞[ that is absolutely
continuous on 0, α and satisfies h(0) = 0 , h(2s) ≤ Kh(s) for all s > 0 , where
K is a constant. Put
Z c
¡√ ¢
γ −(n+2)/2 h γ dγ
F (b, c) = κn
b
whenever 0 < b < c ≤ α .
(i) The set
¾
½
MD (u, p, b) − MD (u, p, c)
=∞
(14)
p ∈ E : lim sup
F (b, c)
0<b<c→0
has P − h -measure zero.
(ii) If, in addition,
then the set
½
Z
√
0
α
s−n−1 h(s) ds = ∞,
MD (u, p, b)
(15)
p ∈ E : lim sup
=∞
F (b, α)
b→0
also has P − h -measure zero.
¾
48
Neil A. Watson
Proof. The sets (14) and (15) are subsets of
¡ 0
¢
¾
µ ΩD (p, d)
(16)
p ∈ E : lim sup
¡√ ¢ = ∞ ,
h d
d→0
¡√ ¢
by Theorem 5 and its Corollary 1 (with f (γ) = h γ ). Furthermore, since
GD ≤ G on D (by [7, Theorem 10]), we have ΩD (p, d) ⊆ Ω(p, d) . Therefore,
p
0
given d > 0 , if we choose r = 3 nd/e then ΩD (p, d) ⊆ P (p, r) . It follows that
the set (16) is a subset of
¡
¢
¾
½
µ P (p, r)
=∞ ,
(17)
p ∈ E : lim sup
h(δr)
r→0
p
where δ = e/9n < 1 . If i is chosen so that 2i δ > 1 , then
½
h(δr) ≥ K −i h(2i δr) ≥ K −i h(r).
Therefore the set (17) is a subset of
¡
¢
¾
½
µ P (p, r)
=∞ ,
p ∈ E : lim sup
h(r)
r→0
which has P − h -measure zero by the lemma in [11].
Remark. The case D = Rn+1 of Theorem 6(ii) was proved in [11].
Corollary. Let E be an open set, let D be an open superset of E that is
Dirichlet regular for the adjoint operator, and let u be a supertemperature on E .
(i) If 0 < β < n , then the set
½
¾
MD (u, p, b) − MD (u, p, c)
p ∈ E : lim sup
=∞
b−(n−β)/2 − c−(n−β)/2
0<b<c→0
has P − Λβ -measure zero.
(ii) The set
½
MD (u, p, b) − MD (u, p, c)
p ∈ E : lim sup
=∞
log(c/b)
0<b<c→0
¾
has P − Λn -measure zero.
(iii) If n < β ≤ n + 2 , then the set
½
¾
MD (u, p, b) − MD (u, p, c)
(18)
p ∈ E : lim sup
=∞
c(β−n)/2 − b(β−n)/2
0<b<c→0
has P − Λβ -measure zero.
A generalized Nevanlinna theorem for supertemperatures
49
Proof. The result is obtained by taking h(s) = sβ for β ∈]0, n + 2] in
Theorem 6. (Note that P −Λβ −m(S) = 0 for all subsets S of Rn+1 if β > n+2 .)
Remark. In Theorem 6(ii), the extra condition on h ensures that F (b, α) →
∞ as b → 0 , so that u(p) = ∞ for all p in the set (15); thus the set is polar. In
Theorem 6(i), polarity is not so readily determined, and whether or not the set
(14) is polar depends upon h . By [5, Theorem 1], if a set is not polar then its
P − Λn -measure must be strictly positive, and so the sets considered in parts (i)
and (ii) of the corollary are polar. The set in part (iii), however, may not be polar,
as the following example shows.
1
3
Example. Given β suchQ
that n < β ≤ n + 2 , put α = 31 (n + 3 − β) , so that
n
≤ α < 1 . Let t0 ∈ R , Q = i=1 ]0, 1[ , S = Q×]t0 − 1, t0 [ , and
µ(B) =
ZZ
B∩S
(t0 − t)−α dx dt
for every Borel subset B of Rn+1 . Then µ is a finite measure, so that Gµ is a
positive supertemperature on Rn+1 . We consider the case D = E = Rn+1 of the
above corollary. Given any x0 ∈ Q , we put p0 = (x0 , t0 ) and can find c0 > 0 such
that Ω(p0 , c0 ) ⊆ S . Then, whenever c < c0 , we have
¢
¡ 0
µ Ω (p0 , c) =
ZZ
Ω(p0 ,c)
(t0 − t)−α dx dt = Cn,α c(n+2−2α)/2 ,
where Cn,α = 2n+1 (πn)n/2 (n + 2 − α)−(n+2)/2 . Therefore
because
¡ 0
¢
c−β/2 µ Ω (p0 , c) = Cn,α c(n+2−β−2α)/2 → ∞
as c → 0,
n−β
n + 2 − β − 2α
=
< 0.
2
6
Therefore, by Theorem 5, Corollary 2,
M (u, p0 , b) − M (u, p0 , c)
→∞
c(β−n)/2 − b(β−n)/2
as 0 < b < c → 0 , so that the set (18) contains Q × {t0 } , which is not polar.
50
Neil A. Watson
5. The Riesz measures of supertemperatures on Dirichlet regular sets
In this section we generalize [11, Theorem 5] from the case of a lower halfspace Rn ×] − ∞, a[ to that of an arbitrary open set which is Dirichlet regular for
the adjoint operator.
Theorem 7. Let D be an open
© set which ªis Dirichlet regular for the adjoint
heat operator, let p0 ∈ D , let E ∈ D, Λ(p0 , D) , and let µ be a positive measure
on E .
(i) If µ is the Riesz measure of a supertemperature which has a thermic minorant
on E , then
Z ∞
¢
¡ 0
(19)
γ −(n+2)/2 µ ΩD (p, γ) dγ < ∞
1
for all p ∈ E .
(ii) Conversely, if there is p ∈ E such that (19) holds, then GD µ is a supertemperature on Λ(p, D) . If, in addition,
Z
1
0
then GD µ(p) < ∞ .
¢
¡ 0
γ −(n+2)/2 µ ΩD (p, γ) dγ < ∞,
Proof. By a result in [9], the domain Λ(p0 , D) is Dirichlet regular for the
adjoint heat operator. Furthermore, the Green function for Λ(p0 , D) is the restriction
of G¢D to Λ(p0 , D) × Λ(p0 , D) (by [7, Theorem 14], or [3, p. 300]), and
¡
Λ p, Λ(p0 , D) = Λ(p, D) for any p ∈ Λ(p0 , D) . It therefore suffices to prove the
result when E = D .
(i) Let w be a supertemperature which has a thermic minorant u on D , and
whose Riesz measure is µ . Then µ is also the Riesz measure for w − u . Therefore,
if p ∈ D and c , d are regular values such that c < d , Theorem 1 shows that
Z d
¢
¡ 0
MD (w − u, p, c) = MD (w − u, p, d) + κn
γ −(n+2)/2 µ ΩD (p, γ) dγ
≥ κn
Z
c
d
c
¢
¡ 0
γ −(n+2)/2 µ ΩD (p, γ) dγ.
Making d → ∞ we obtain (19), because MD (w − u, p, c) < ∞ .
(ii) Suppose that (19) holds for some p = p1 ∈ D . Let {kj } be an unbounded
increasing sequence of regular values, and put
0
AD (p1 ; k1 , ∞) = Λ(p1 , D)\ΩD (p1 , k1 ),
0
AD (p1 ; k1 , kj ) = ΩD (p1 , kj )\ΩD (p1 , k1 )
for j > 1.
A generalized Nevanlinna theorem for supertemperatures
51
If u = GD µ , then for all p ∈ D we put
u(p) =
Z
GD (p, q) dµ(q) +
0
ΩD (p1 ,k1 )
Z
GD (p, q) dµ(q)
AD (p1 ;k1 ,∞)
= v1 (p) + v2 (p),
say, and
Z
uj (p) =
GD (p, q) dµ(q)
AD (p1 ;k1 ,kj )
for j ≥ 1 . Since µ is locally finite, v1 and every uj is a supertemperature on D .
Since {uj } is increasing to the limit v2 , v2 is a supertemperature on Λ(p1 , D) if
¢
¡ 0
v2 (p1 ) < ∞ , by [6, Theorem 6]. Writing λ(γ) = µ ΩD (p1 , γ) for all γ > 0 , we
have
Z kj
Z kj
£
¤kj
τ 0 (γ)λ(γ) dγ.
τ (γ) dλ(γ) = τ (γ)λ(γ) k −
uj (p1 ) =
1
k1
k1
Since (19) holds when p = p1 , we have
λ(c)τ (c) = λ(c)
Z
∞
c
0
−τ (γ) dγ ≤ −
Z
as c → ∞ . Therefore
v2 (p1 ) = lim uj (p1 ) = −τ (k1 )λ(k1 ) −
j→∞
∞
c
Z
τ 0 (γ)λ(γ) dγ → 0
∞
k1
τ 0 (γ)λ(γ) dγ < ∞,
so that v2 , and hence u , is a supertemperature on Λ(p1 , D) .
For the last part, let {cj } be a decreasing null sequence of regular values
(relative to p1 ). Then
u(p1 ) = lim
j→∞
= lim
j→∞
≤−
Z
Z
∞
µ
−τ (cj )λ(cj ) −
∞
0
τ (γ) dλ(γ)
cj
Z
∞
cj
τ 0 (γ)λ(γ) dγ < ∞.
0
τ (γ)λ(γ) dγ
¶
52
Neil A. Watson
6. Differences of positive supertemperatures on Λ(p0 , D)
Let u be a δ -subtemperature on an open set E , and let D be an open
superset of E that is Dirichlet regular for the adjoint operator. If µ is the Riesz
measure for u , then µ can be written minimally as a difference µ+ − µ− of two
positive measures on E . Whenever ΩD (p, c) ⊆ E , we put
Z c
¡ 0
¢
+
+
+
λD (p, c) = µ ΩD (p, c) ,
ND (p, c) = −
τ 0 (γ)λ+
D (p, γ) dγ,
0
+
−
and similarly for µ− . We say that u(p0 ) is finite if ND
(p0 , · ) and ND
(p0 , · ) are
both finite-valued, in which case it follows from (2) that u is the difference of
two supertemperatures which are finite at p0 . If u(p0 ) is finite, we define the
characteristic TD of u at p0 by
+
TD (u, p0 , c) = MD (u+ , p0 , c) + ND
(p0 , c) − u(p0 )
for each regular value of c such that ΩD (p0 , c) ⊆ E . We can use TD to characterize
those δ -subtemperatures on Λ(p0 , D) or D that can be written as a difference of
two positive supertemperatures, and thus generalize [11, Theorem 6].
Theorem 8. Let D be an open set which
© is Dirichlet
ª regular for the adjoint heat operator, let p0 ∈ D , let E ∈ D, Λ(p0 , D) , and let u be a δ subtemperature on E .
(i) If u = u1 − u2 is the difference of two positive supertemperatures on E ,
and u(p1 ) is finite, then TD (u, p1 , · ) is an increasing function such that 0 ≤
TD (u, p1 , c) ≤ u2 (p1 ) for all regular values of c , and there is a convex function
φ such that TD (u, p1 , · ) = φ ◦ τ .
S∞
(ii) Conversely, if there is a sequence {pj } in E such that E = j=1 Λ(pj , D) ,
u(pj ) is finite for all j , and TD (u, pj , · ) is bounded above on the set of all
regular values for each j , then u is the difference of two positive supertemperatures on E .
Proof. (i) For i ∈ {1, 2} , let µi be the Riesz measure for ui , and put
Z c
¡ 0
¢
i
i
λD (p1 , c) = µi ΩD (p1 , c) ,
ND (p1 , c) = −
τ 0 (γ)λiD (p1 , γ) dγ
0
for all c > 0 . Since u1 ≥ 0 , it follows from (2) that
1
0 = MD (u−
1 , p1 , c) = MD (u1 , p1 , c) + ND (p1 , c) − u1 (p1 ).
Since µ1 and µ2 are positive and µ = µ1 − µ2 , we have µ+ ≤ µ1 and µ− ≤ µ2 ,
so that
+
1
ND
(p1 , c) ≤ ND
(p1 , c) = u1 (p1 ) − MD (u1 , p1 , c).
A generalized Nevanlinna theorem for supertemperatures
53
Furthermore u1 ≥ u+ , so that MD (u+ , p1 , c) ≤ MD (u1 , p1 , c) . Hence
¡
¢
TD (u, p1 , c) ≤ MD (u1 , p1 , c) + u1 (p1 ) − MD (u1 , p1 , c) − u(p1 ) = u2 (p1 ).
Now let v2 = GD µ− and v1 = u + v2 . Applying (2) to each vi and subtracting,
we obtain
+
−
u(p1 ) = MD (u, p1 , c) + ND
(p1 , c) − ND
(p1 , c),
so that
−
TD (u, p1 , c) = MD (u+ , p1 , c) + ND
(p1 , c) − MD (u, p1 , c)
−
= MD (u− , p1 , c) + ND (p1 , c)
= MD (u− , p1 , c) + v2 (p1 ) − MD (v2 , p1 , c)
= v2 (p1 ) − MD (v2 − u− , p1 , c).
Let p ∈ E . If u(p) ≥ 0 , then v1 (p) ≥ v2 (p) and v2 (p) − u− (p) = v2 (p) =
(v1 ∧v2 )(p) . On the other hand, if u(p) < 0 then v1 (p) < v2 (p) and v2 (p)−u− (p) =
v1 (p) = (v1 ∧ v2 )(p) . Hence
TD (u, p1 , c) = v2 (p1 ) − MD (v1 ∧ v2 , p1 , c).
Since v1 ∧ v2 is a supertemperature on E , the characteristic TD (u, p1 , · ) is increasing and real-valued on the set of regular values (by [10, Theorem 2]), there
is a convex function φ such that TD (u, p1 , · ) = φ ◦ τ (by [10, Theorem 3]) and
TD (u, p1 , 0+) = v2 (p1 ) − (v1 ∧ v2 )(p1 ) ≥ 0 (by [10, Theorem 2]).
(ii) The proof is similar to that of [11, Theorem 6(ii)].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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Brzezina, M.: A note on the convexity theorem for mean values of subtemperatures. Ann. Acad. Sci. Fenn. Math. 21, 1996, 111–115.
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Sternberg, S.: Lectures on Differential Geometry. - Prentice-Hall, Englewood Cliffs,
1964.
Taylor, S.J., and N.A. Watson: A Hausdorff measure classification of polar sets for
the heat equation. - Math. Proc. Cambridge Philos. Soc. 97, 1985, 325–344.
Watson, N.A.: A theory of subtemperatures in several variables. - Proc. London Math.
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Watson, N.A.: Green functions, potentials, and the Dirichlet problem for the heat equation. - Proc. London Math. Soc. 33, 1976, 251–298.
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[10]
[11]
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Watson, N.A.: Mean values of subtemperatures over level surfaces of Green functions. Ark. Mat. 30, 1992, 165–185.
Watson, N.A.: Nevanlinna’s first fundamental theorem for supertemperatures. - Math.
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Received 22 November 2000